Prove that the product of two consecutive odd numbers is equal to the square of the even number between them minus 1.

1. Problem Description

Prove that the product of two consecutive odd numbers is equal to the square of the even number between them minus

1.

2. Solution Steps

Let the two consecutive odd numbers be $2n-1$ and $2n+1$, where $n$ is an integer. The even number between them is $2n$.

The product of the two consecutive odd numbers is:

$(2n-1)(2n+1) = 4n^2 - 1$

The square of the even number between them is $(2n)^2 = 4n^2$.

Subtracting 1 from the square of the even number gives $4n^2 - 1$.

Since $(2n-1)(2n+1) = 4n^2 - 1$ and $(2n)^2 - 1 = 4n^2 - 1$, we have shown that the product of two consecutive odd numbers is equal to the square of the even number between them minus

1.

3. Final Answer

The product of two consecutive odd numbers is equal to the square of the even number between them minus

1.